Short Notes: Energy Formulation | Short Notes for Mechanical Engineering PDF Download

 Page 1


Energy Formulation
Torque and Angular Acceleration of a Rigid Body 
For a rigid body, net torque acting
t = la
Where, a = angular acceleration of rigid body 
/ = moment of inertia about axis of rotation
• Kinetic energy of a rigid body rotating about fixed axis
K E - - l a 1 ( a = mgular velocity')
• Angular moment of a particle about same point
L = r x p 
L = m(r x v)
o----- ?
p = mv
Angular moment
Where L = angular displacement
• Angular moment of a rigid body rotating about a fixed axis.
Page 2


Energy Formulation
Torque and Angular Acceleration of a Rigid Body 
For a rigid body, net torque acting
t = la
Where, a = angular acceleration of rigid body 
/ = moment of inertia about axis of rotation
• Kinetic energy of a rigid body rotating about fixed axis
K E - - l a 1 ( a = mgular velocity')
• Angular moment of a particle about same point
L = r x p 
L = m(r x v)
o----- ?
p = mv
Angular moment
Where L = angular displacement
• Angular moment of a rigid body rotating about a fixed axis.
s
Angular moment of a rigid body
• Angular moment of a rigid body in combined rotation and translation
L = i-cM + M(r0xv0 )
Combined rotation and translation in a rigid body 
• Conservation of angular momentum
_ —
~ ~ d r
dL
— = r x F +vx p 
dt y
• Kinetic energy of rigid body in combined translational and rotational motion.
1. Kinetic energy associated with the motion of the centre of mass CM of the 
body as if the total mass were concentrated at that point.
2. Kinetic energy associated with the rotation of the body about an appropriate 
axis through CM.
Sum of (i) and (ii) is stated as kinetic energy of rigid body.
_. 1 2 1 T 2
K = — m vcv + — I C if a
Uniform Pure Rolling
Pure rolling means no relative motion or no slipping at point of contact between 
two bodies.
If Vp = VQ=* no slipping 
v = Ru)
If Vp >V q = > forward slipping 
v> Rc j
If Vp < VQ= > backward slipping
v< Rcj
Page 3


Energy Formulation
Torque and Angular Acceleration of a Rigid Body 
For a rigid body, net torque acting
t = la
Where, a = angular acceleration of rigid body 
/ = moment of inertia about axis of rotation
• Kinetic energy of a rigid body rotating about fixed axis
K E - - l a 1 ( a = mgular velocity')
• Angular moment of a particle about same point
L = r x p 
L = m(r x v)
o----- ?
p = mv
Angular moment
Where L = angular displacement
• Angular moment of a rigid body rotating about a fixed axis.
s
Angular moment of a rigid body
• Angular moment of a rigid body in combined rotation and translation
L = i-cM + M(r0xv0 )
Combined rotation and translation in a rigid body 
• Conservation of angular momentum
_ —
~ ~ d r
dL
— = r x F +vx p 
dt y
• Kinetic energy of rigid body in combined translational and rotational motion.
1. Kinetic energy associated with the motion of the centre of mass CM of the 
body as if the total mass were concentrated at that point.
2. Kinetic energy associated with the rotation of the body about an appropriate 
axis through CM.
Sum of (i) and (ii) is stated as kinetic energy of rigid body.
_. 1 2 1 T 2
K = — m vcv + — I C if a
Uniform Pure Rolling
Pure rolling means no relative motion or no slipping at point of contact between 
two bodies.
If Vp = VQ=* no slipping 
v = Ru)
If Vp >V q = > forward slipping 
v> Rc j
If Vp < VQ= > backward slipping
v< Rcj
Pure Rolling
No slippig s = 2 t t R 
Forward slipping s > 2t t R 
Backward slipping s < 2 t t R 
Accelerated Pure Rolling
A pure rolling is equivalent to pure translation and pure rotation. It follows a 
uniform rolling and accelerated pure rolling can be defined as 
F + f = Ma 
(F - f).R = la
Accelerated pure rolling
F = force them acting on a body 
f = friction on that body 
Angular Impulse
The angular Impulse of a torque in a given time Interval is defined as
where L2 and Li are the angular momentum at time f2 and fi respectively.
Collision: A Collision Is an Isolated event in which two or more moving bodies exert 
forces on each other for a relatively short time. Collision between two bodies may 
be classified in two ways: Head-on collision, and Oblique collision.
Heat-on Collision: Let the two balls of masses mi and m2 collide directly with each 
other with velocities Vi and v2 in direction as shown in figure. After collision the 
velocity become
v,
and
along the same line.
Before collision After collision
Page 4


Energy Formulation
Torque and Angular Acceleration of a Rigid Body 
For a rigid body, net torque acting
t = la
Where, a = angular acceleration of rigid body 
/ = moment of inertia about axis of rotation
• Kinetic energy of a rigid body rotating about fixed axis
K E - - l a 1 ( a = mgular velocity')
• Angular moment of a particle about same point
L = r x p 
L = m(r x v)
o----- ?
p = mv
Angular moment
Where L = angular displacement
• Angular moment of a rigid body rotating about a fixed axis.
s
Angular moment of a rigid body
• Angular moment of a rigid body in combined rotation and translation
L = i-cM + M(r0xv0 )
Combined rotation and translation in a rigid body 
• Conservation of angular momentum
_ —
~ ~ d r
dL
— = r x F +vx p 
dt y
• Kinetic energy of rigid body in combined translational and rotational motion.
1. Kinetic energy associated with the motion of the centre of mass CM of the 
body as if the total mass were concentrated at that point.
2. Kinetic energy associated with the rotation of the body about an appropriate 
axis through CM.
Sum of (i) and (ii) is stated as kinetic energy of rigid body.
_. 1 2 1 T 2
K = — m vcv + — I C if a
Uniform Pure Rolling
Pure rolling means no relative motion or no slipping at point of contact between 
two bodies.
If Vp = VQ=* no slipping 
v = Ru)
If Vp >V q = > forward slipping 
v> Rc j
If Vp < VQ= > backward slipping
v< Rcj
Pure Rolling
No slippig s = 2 t t R 
Forward slipping s > 2t t R 
Backward slipping s < 2 t t R 
Accelerated Pure Rolling
A pure rolling is equivalent to pure translation and pure rotation. It follows a 
uniform rolling and accelerated pure rolling can be defined as 
F + f = Ma 
(F - f).R = la
Accelerated pure rolling
F = force them acting on a body 
f = friction on that body 
Angular Impulse
The angular Impulse of a torque in a given time Interval is defined as
where L2 and Li are the angular momentum at time f2 and fi respectively.
Collision: A Collision Is an Isolated event in which two or more moving bodies exert 
forces on each other for a relatively short time. Collision between two bodies may 
be classified in two ways: Head-on collision, and Oblique collision.
Heat-on Collision: Let the two balls of masses mi and m2 collide directly with each 
other with velocities Vi and v2 in direction as shown in figure. After collision the 
velocity become
v,
and
along the same line.
Before collision After collision
V,
W j - em2' m, 4- em 2
m : — m 2
v i +
m l 4- m2
m i ~ em2
| m l + enu
W j + m 2
{ m \ + m 2 .
Where, mi = mass of body 1 
m2 = mass of body 2 
Vi = velocity of body 1 
v2 = velocity of body 2
v ;
= velocity of body 1 after collision
vj
= velocity of body 2 after collision 
Where e = coefficient restitution
Separation speed 
Approach speed
v2 ~ Vj
• In case of head-on elastic collision
e = 1
• In case of head-on inelastic collision
0 < e < 1
• In case of head-on perfectly inelastic collision
e = 0
If e is coefficient of restitution between ball and ground, then after nth collision 
with the floor, the speed of ball will remain envg and it will go upto a height e2 n h.
\ - e ‘n h
.Ou = o
_L°
+
u0 =1 2 gh 
Collision of a ball 
with floor
Oblique Collision: In case of oblique collision linear momentum of individual 
particle do change along the common normal direction. No component of impulse 
act along common tangent direction. So, linear momentum or linear velocity 
remains unchanged along tangential direction. Net momentum of both the particle 
remains conserved before and after collision in any direction.
Page 5


Energy Formulation
Torque and Angular Acceleration of a Rigid Body 
For a rigid body, net torque acting
t = la
Where, a = angular acceleration of rigid body 
/ = moment of inertia about axis of rotation
• Kinetic energy of a rigid body rotating about fixed axis
K E - - l a 1 ( a = mgular velocity')
• Angular moment of a particle about same point
L = r x p 
L = m(r x v)
o----- ?
p = mv
Angular moment
Where L = angular displacement
• Angular moment of a rigid body rotating about a fixed axis.
s
Angular moment of a rigid body
• Angular moment of a rigid body in combined rotation and translation
L = i-cM + M(r0xv0 )
Combined rotation and translation in a rigid body 
• Conservation of angular momentum
_ —
~ ~ d r
dL
— = r x F +vx p 
dt y
• Kinetic energy of rigid body in combined translational and rotational motion.
1. Kinetic energy associated with the motion of the centre of mass CM of the 
body as if the total mass were concentrated at that point.
2. Kinetic energy associated with the rotation of the body about an appropriate 
axis through CM.
Sum of (i) and (ii) is stated as kinetic energy of rigid body.
_. 1 2 1 T 2
K = — m vcv + — I C if a
Uniform Pure Rolling
Pure rolling means no relative motion or no slipping at point of contact between 
two bodies.
If Vp = VQ=* no slipping 
v = Ru)
If Vp >V q = > forward slipping 
v> Rc j
If Vp < VQ= > backward slipping
v< Rcj
Pure Rolling
No slippig s = 2 t t R 
Forward slipping s > 2t t R 
Backward slipping s < 2 t t R 
Accelerated Pure Rolling
A pure rolling is equivalent to pure translation and pure rotation. It follows a 
uniform rolling and accelerated pure rolling can be defined as 
F + f = Ma 
(F - f).R = la
Accelerated pure rolling
F = force them acting on a body 
f = friction on that body 
Angular Impulse
The angular Impulse of a torque in a given time Interval is defined as
where L2 and Li are the angular momentum at time f2 and fi respectively.
Collision: A Collision Is an Isolated event in which two or more moving bodies exert 
forces on each other for a relatively short time. Collision between two bodies may 
be classified in two ways: Head-on collision, and Oblique collision.
Heat-on Collision: Let the two balls of masses mi and m2 collide directly with each 
other with velocities Vi and v2 in direction as shown in figure. After collision the 
velocity become
v,
and
along the same line.
Before collision After collision
V,
W j - em2' m, 4- em 2
m : — m 2
v i +
m l 4- m2
m i ~ em2
| m l + enu
W j + m 2
{ m \ + m 2 .
Where, mi = mass of body 1 
m2 = mass of body 2 
Vi = velocity of body 1 
v2 = velocity of body 2
v ;
= velocity of body 1 after collision
vj
= velocity of body 2 after collision 
Where e = coefficient restitution
Separation speed 
Approach speed
v2 ~ Vj
• In case of head-on elastic collision
e = 1
• In case of head-on inelastic collision
0 < e < 1
• In case of head-on perfectly inelastic collision
e = 0
If e is coefficient of restitution between ball and ground, then after nth collision 
with the floor, the speed of ball will remain envg and it will go upto a height e2 n h.
\ - e ‘n h
.Ou = o
_L°
+
u0 =1 2 gh 
Collision of a ball 
with floor
Oblique Collision: In case of oblique collision linear momentum of individual 
particle do change along the common normal direction. No component of impulse 
act along common tangent direction. So, linear momentum or linear velocity 
remains unchanged along tangential direction. Net momentum of both the particle 
remains conserved before and after collision in any direction.